https://doi.org/10.1007/s100510070126
Exact Bures probabilities that two quantum bits are classically correlated
ISBER, University of California, Santa Barbara, CA 93106-2150, USA
Received:
10
December
1999
Revised:
24
February
2000
Published online: 15 October 2000
In previous studies, we have explored the ansatz that the volume
elements of the Bures metrics over
quantum systems
might serve as prior distributions, in analogy with the
(classical) Bayesian role of the volume elements ("Jeffreys' priors")
of Fisher information metrics.
Continuing this work, we obtain exact Bures prior
probabilities that the members of
certain low-dimensional subsets of the fifteen-dimensional
convex set of
density matrices are separable
or classically
correlated.
The main analytical tools employed are symbolic integration and
a formula of Dittmann (J. Phys. A 32, 2663 (1999))
for Bures metric
tensors.
This study complements an earlier one (J. Phys. A 32,
5261 (1999)) in which numerical
(randomization) -but not integration -methods
were used to estimate Bures separability probabilities for
unrestricted
and
density matrices.
The exact values adduced here for pairs of quantum bits (qubits),
typically, well exceed the estimate (
) there, but
this disparity may be attributable to our
focus on special low-dimensional subsets.
Quite remarkably, for
the q= 1 and
states inferred using the
principle of maximum nonadditive (Tsallis)
entropy, the Bures probabilities
of separability are both equal to
.
For the Werner qubit-qutrit
and qutrit-qutrit
states, the probabilities
are vanishingly small, while in the qubit-qubit case it is
.
PACS: 03.67.-a – Quantum information / 03.65.Bz – Foundations, theory of measurement, miscellaneous theories / 02.40.Ky – Riemannian geometries / 02.50.-r – Probability theory, stochastic processes, and statistics
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000